Principia Mathematica

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For Isaac Newton's 1687 book containing basic laws of physics, see Philosophiae Naturalis Principia Mathematica.

The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. It is an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic. One of the main inspirations and motivations for the Principia was Frege's earlier work on logic, which had led to some contradictions discovered by Russell. These were avoided in the Principia by building an elaborate system of types: a set has a higher type than its elements and one can not speak of the "set of all sets" and similar constructs which lead to paradoxes (see Russell's paradox).

The Principia only covered set theory, cardinal numbers, ordinal numbers and real numbers; deeper theorems from real analysis were not included, but by the end of the third volume it was clear that all known mathematics could in principle be developed in the adopted formalism.

The questions remained whether a contradiction could be derived from the Principia's axioms, and whether there exists a mathematical statement which could neither be proven nor disproven in the system. These questions were settled by Gödel's incompleteness theorem in 1931. Gödel's second incompleteness theorem shows that basic arithmetic cannot be used to prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger. In other words, the statement "there are no contradictions in the Principia system" cannot be proven true or false in the Principia system unless there are contradictions in the system (in which case it can be proven both true and false).

Yet, as Douglas Hofstadter has pointed out, there may be additional levels of potential contradiction here. A central principle of the "system of types" mentioned above is that statements that are self referential are forbidden, to avoid Russell's paradox. Loops of statements that are self referential (circular definitions) are also forbidden. However, the statement "We do not allow self-referential statements in Principia Mathematica" is a seeming violation of the rule against self-referential statements, an apparent contradiction at the heart of the philosophy, although it may be interpreted as meaning that none of the following statements in the formal system itself would be self-referential. That is, this statement may mean "in the following formal axiomatic system self-referential statements are not allowed," which clearly is not self-referential.

A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third.

The Principia is widely considered by specialists in the subject to be one of the most important and seminal works in mathematical logic and philosophy.

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